I have so many things to write to you about my new discoveries, that I
cannot solve it in another way, than doing nothing else than
writing you this letter; without waiting for your answer to my previous letter;
and perhaps I shouldn't write to you before getting it, but
I will put my letter addressed to the Baroness in My Father's envelope
asking you the favour of handing it her personally.
First of all I do answer the Binomial Theorem. For positive integers exponents the
proof, of course is perfect as you have written, but you need to know the form of the series, to be used in the proof,

There follows some discussion of the Binomial Theorem for non-integer exponents. He discusses his father's proof and mentions those of Lacroix (1765 - 1843) and Vega (1754 - 1802).

... About these things we will continue to discuss; we will try it
for negative exponents as well. As far as possible.
I have got already the conditions, and once I put them in order, and finish them, I will
publish a paper about the parallels; right at this moment is not finished, but the way
I followed, is almost sure promising to attain the goal, if it were ever possible;
it is not finished but I have found beautiful things, that surprised even me,
and it would be a pity to lose them; my Dearest Father will see and know;
I cannot say more, only that from nothing I have created a new different world;
Everything I sent you before is like a house of cards if compared to a tower.
I am convinced, that it will be in my honour, not less than if I would discover ...
Waiting for your answer, yours for ever indebted son Bolyai.

He then writes some postscripts to the letter, filling in some of the gaps in the writing including down the left-hand side of the first page.